UFL Specification and User Manual 0.3 - FEniCS Project

UFL Specification and User Manual 0.3Martin S. Alnæs, Anders Loggcell = triangleBDM1 = FiniteElement("Brezzi-Douglas-Marini", cell, 1)DG0 = FiniteElement("Discontinuous Lagrange", cell, 0)element = BDM1 * DG0(tau, w) = TestFunctions(element)(sigma, u) = TrialFunctions(element)f = Function(DG0)a = (dot(tau, sigma) - div(tau)*u + w*div(sigma))*dxL = w*f*dxThis example is implemented in the file mixed poisson.ufl in the collectionof demonstration forms included with the UFL source distribution.3.9 Poisson’s equation with DG elementsWe consider again Poisson’s equation, but now in an (interior penalty) discontinuousGalerkin formulation: Find u ∈ V = L 2 such thata(v,u) = L(v) ∀v ∈ V,where∫a(v,u;h) =Ω+ ∑ S∫+∫L(v;f,g) =∂Ω∇v ·∇udx∫−〈∇v〉·u n − v n ·〈∇u〉+(α/h)v n · u n dSS−∇v · u n − v n ·∇u+(γ/h)vuds∫vf dx+ vgds.Ω ∂Ω(3.22)69